Let us consider the equation of the required circle be

(x – h)^{2 }+ (y – k)^{2} = r^{2}

We know that the circle passes through points (2,3) and (-1,1).

(2 – h)^{2}+ (3 – k)^{2} =r^{2} ……………..(1)

(-1 – h)^{2}+ (1– k)^{2} =r^{2} ………………(2)

Since, the centre (h, k) of the circle lies on line x – 3y – 11= 0,

h – 3k =11………………… (3)

From the equation (1) and (2), we obtain

(2 – h)^{2}+ (3 – k)^{2} =(-1 – h)^{2} + (1 – k)^{2}

4 – 4h + h^{2} +9 -6k +k^{2} = 1 + 2h +h^{2}+1 – 2k + k^{2}

4 – 4h +9 -6k = 1 + 2h + 1 -2k

6h + 4k =11……………. (4)

Now let us multiply equation (3) by 6 and subtract it from equation (4) to get,

6h+ 4k – 6(h-3k) = 11 – 66

6h + 4k – 6h + 18k = 11 – 66

22 k = – 55

K = \( \dfrac{-5}{2}\)

Substitute this value of K in equation (4) to get,

6h + 4(\( \dfrac{-5}{2}\)) = 11

6h – 10 = 11

6h = 21

h = \( \dfrac{21}{6}\)

h = \( \dfrac{7}{2}\)

We obtain h = \( \dfrac{7}{2}\)and k =\( \dfrac{-5}{2}\)

On substituting the values of h and k in equation (1), we get

(2 – \(
\dfrac{7}{2}\))^{2} + (3 + \(
\dfrac{5}{2}\))^{2} = r^{2}

\( (\dfrac{4-7}{2})^2+(\dfrac{6+5}{2})^2=r^2\)

\(
(\dfrac{-3}{2})^2\) + \(
(\dfrac{11}{2})^2\)= r^{2}

\(
\dfrac{9}{4}\) + \(
\dfrac{121}{4}\) = r^{2}

\(
\dfrac{130}{4}\) = r^{2}

The equation of the required circle is

(x – \(
\dfrac{7}{2}\))^{2} + (y + \(
\dfrac{5}{2}\))^{2} = \(
\dfrac{130}{4}\)

\( (\dfrac{2x-7}{2})^2+(\dfrac{2y+5}{2})^2=\dfrac{130}{4}\)

4x^{2} -28x + 49 +4y^{2} + 20y + 25 =130

4x^{2} +4y^{2} -28x + 20y – 56 = 0

4(x^{2} +y^{2} -7x + 5y – 14) = 0

x^{2 }+ y^{2 }– 7x + 5y – 14 = 0

The equation of the required circle is x^{2 }+ y^{2 }– 7x + 5y – 14 = 0

An equilateral triangle is inscribed in the parabola y^{2} = 4ax, where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.

A man running a racecourse notes that the sum of the distances from the two flag posts from him is always 10 m and the distance between the flag posts is 8 m. Find the equation of the posts traced by the man.

Find the area of the triangle formed by the lines joining the vertex of the parabola x^{2} = 12y to the ends of its latus rectum.

A rod of length 12 cm moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point P on the rod, which is 3 cm from the end in contact with the x-axis.

An arch is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the centre. Find the height of the arch at a point 1.5 m from one end.